Find a power series representation for the function; find the interval of convergence. (Give your power series representation centered at x = 0.)

Question

Find a power series representation for the function; find the interval of convergence. (Give your power series representation centered at x = 0.)

f(x)= ((x^2)+1)/(2x-1)

[?] + sum from n=0 to infinity [?] provided |x| < [?]

Answer 1

I am NOT certain about this..but try it.

(x² + 1)/(2x - 1) = (x/2) + (1/4) + 5/(8x -4)

5/(8x-4) = -(5/4)/(1 - 2x) =- (5/4)(1 + 2x + 4x² + 8x³ + ...)

You can collect terms to write the power series which will mean that the first two terms will not conform to the pattern of the rest.

Convergence will be determined by the higher order terms...and that series will only converge for |x|<1/2.

If you get a better answer, I would appreciate knowing what it is. Tutors learn too!